Marc MAGRO

Integrability in Gauge and String Theory 2012 - ETH Zürich

1204.0766, 1204.2531, 1206.6050

With F. DELDUC (ENS Lyon) and B. VICEDO (York / Hertfordshire)

Plan

Part I: - Motivation- Non-ultralocality- Difficulties

Part II: Review of first steps of Faddeev-Reshetikhin procedure

Part III: Generalization to the superstring- Alleviation of non-ultralocality- Link with Pohlmeyer reduction

Part IV: Remarks and conclusion

Motivation

Construct corresponding Quantum Integrable Lattice Model

- Long-term goal...- Would provide a proof of quantum integrability

Motivation & Difficulties

Construct corresponding Quantum Integrable Lattice Model

- Necessary Step = Classical integrable discretization

Already very difficult !

Old problem for integrable Sigma Models !

Difficulty comes from their Non-ultralocality

Review: - What is Non-ultralocality - Why it is the origin of difficulties

- Lattice

N sites with periodic conditions

Classical integrable discretization

[Freidel-Maillet '91]Freidel-Maillet Quadratic Algebra

Freidel-Maillet Quadratic Algebra

Antisymmetry

Freidel-Maillet Quadratic Algebra

Jacobi identity

Monodromy has Poisson bracket:

Integrability

Freidel-Maillet Quadratic Algebra

Non-ultralocality

Consider b=0: - Previous conditions imply c=b=0 and a=d with

a solution of modified classical Yang-Baxter equation

- Corresponds to ultralocal model and in particular

Comes from b and c

Continuum Limit

- PB of Lax matrix are of the r/s form with r and s stemming from (a,b,c,d)- Non-ultralocality carried by the matrix s [Maillet '85 '86]

Spatial Lax Matrix

1. Start from continuum2. Hamiltonian Lax matrix known 3. Its PB are of the r/s form

First difficulty for the superstring

[M.M. '08, Vicedo '09]

Good news

Same situation for Principal Chiral Model, Coset Models !

r and s do not stem from any (a,b,c,d)

First difficulty for the superstring

Bad news

r/s algebra of the continuumcannot be discretized

as a lattice Freidel-Maillet algebra

More difficultiesCould say:

1. Start from known PB algebra of the Lax matrix on continuum

2. Compute then PB of

Problem: These PB are not well defined

Schematically, comes from:

Old problem ! What to do ?

Characteristic function of the interval: Undefined when two points coincide !

Faddeev-Reshetikhin approach [FR '86]

Concerns SU(2) Principal Chiral Model

FR Strategy = To get rid of Non-ultralocality

Satisifies a non-ultralocal r/s algebra

First steps of FR approach

First steps of FR approach

Degeneracy of ultralocal bracket

1. Only possible to reproduce Reduction of PCM dynamics defined by setting Casimirs to constants

2. Can be done in a consistent way because these quantities are chiral/antichiral

Reduction of conformal symmetry

Hamiltonian H' for reduced dynamics

How to generalize FR approach to the superstring ?

- Keep Lax matrix

Impossible to guess !

Try to mimick FR:

But- No sign of Coset

- Its Casimirs are inconsistent with the dynamics of the superstring !

How to generalize FR approach to the superstring ?

The way out of the tunnel

Needs to understand better the algebraic setting behind Hamiltonian integrability of the superstring

and more precisely, the deep origin of Non-ultralocality.

Rephrase integrability in the right framework

= R-matrix approach

Already done by B. Vicedo in 1003.1192

[Semenov-Tian-Shansky '83]

Quartet behind integrability

Quartet behind integrability

Origin of non-ultralocality

Origin of non-ultralocality

First try

Impossible to completely get rid of non-ultralocality !

Good lead: Can recast original FR procedure in this language

Second try

r and s stem from (a,b,c,d) satisfying Freidel-Maillet

conditions !

[Semenov-Tian-Shansky and Sevostyanov '95]

Alleviation of non-ultralocality

r and s stem from (a,b,c,d) satisfying Freidel-Maillet

conditions !

[Semenov-Tian-Shansky and Sevostyanov '95]

Alleviation of non-ultralocalityResulting non-ultralocality is mild

Modified PB for phase space variables

Casimirs of the modified Poisson bracketCase of Sigma Model on symmetric space F/G

Casimirs of the modified Poisson bracketCase of Sigma Model on symmetric space F/G

Casimirs and Pohlmeyer reductionCasimirs are consistent with dynamics of Sigma Model !

[Pohlmeyer '76]

Result of the procedure

Non-ultralocality of symmetric space sine-Gordon models viewed as gauged WZW models + Potential is mild

Pohlmeyer reduction of Sigma model on symmetric space F/G = SSSG model

Consequence for symmetric space sine-Gordon (SSSG) models

[Eichenherr, Pohlmeyer '79]

[Bakas et al. '96,Grigoriev and Tseytlin '07]

Case of the superstring

[Grigoriev and Tseytlin '07, Mikhailov and Schäfer-Nameki '07]

Remarks

Pohlmeyer reduction[Mikhailov '05, '06, Schmiddt '10, '11]

Remarks

[Dorey and Hollowood '95, Hoare and Tseytlin '10]

[Maillet '86]

ConclusionNon-ultralocality of generalized sine-Gordon models is mild

Challenge: Reach same situation as for ultralocal models. One knows from [Freidel-Maillet '91 '92] that one has to search for

representation of Quantum Algebra

Generalization of first steps of FR procedure=

Pohlmeyer reduction

Conclusion

Appealing structure which brings hope that one may be able to quantize from first principles

at least the Pohlmeyer reduction of the superstring !